Disconjugacy and disfocality criteria for second order singular half-linear differential equations

• Autorzy: Ondřej Došlý , Alexander Lomtatidze
• Języki publikacji: EN

Abstrakty

EN

We establish Vallée Poussin type disconjugacy and disfocality criteria for the half-linear second order differential equation
$u'' = p(t)|u|^α |u'|^{1-α} sgn u + g(t)u'$,
where α ∈ (0,1] and the functions $p,g ∈ L_{loc}(a,b)$ are allowed to have singularities at the end points t = a, t = b of the interval under consideration.

Słowa kluczowe

EN

Vallée Poussin type criterion; half-linear equation; disfocality; disconjugacy

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 72
• Numer: 3
• Strony: 273-284

Daty

wydano

1999

otrzymano

1999-05-12

Twórcy

autor

Ondřej Došlý

• Mathematical Institute, Czech Academy of Sciences, Žižkova 22, CZ-616 62 Brno, Czech Republic

autor

Alexander Lomtatidze

• Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, CZ-662 95 Brno, Czech Republic

Bibliografia

• [1] J. Andres, On the criterion of de la Vallée Poussin, Publ. Math. Debrecen 45 (1994), 145-152.
• [2] T. Chantladze, N. Kandelaki and A. Lomtatidze, On zeros of solutions of second order singular half-linear equations, Mem. Differential Equations Math. Phys. 17 (1999), 127-154.
• [3] J. H. E. Cohn, On an oscillation criterion of de la Vallée Poussin, Quart. J. Math. Oxford Ser. (2) 39 (1988), 173-174.
• [4] O. Došlý, A remark on conjugacy of half-linear second order differential equations, Math. Slovaca, to appear.
• [5] O. Došlý, Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J. 28 (1998), 507-521.
• [6] A. Elbert and T. Kusano, Principal solutions of nonoscillatory half-linear differential equations, Adv. Math. Sci. Appl. 18 (1998), 745-759.
• [7] A. Elbert, T. Kusano and T. Tanigawa, An oscillatory half-linear differential equation, Arch. Math. (Brno) 33 (1997), 165-172.
• [8] B. J. Harris, On an oscillation criterion of Cohn, Quart. J. Math. Oxford Ser. (2) 42 (1991), 309-313.
• [9] P. Hartman and A. Wintner, On an oscillation criterion of de la Vallée Poussin, Quart. Appl. Math. 13 (1955), 330-332.
• [10] I. T. Kiguradze and A. G. Lomtatidze, On certain boundary value problems for second order linear ordinary differential equations with singularities, J. Math. Anal. Appl. 101 (1984), 325-347.
• [11] I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second order ordinary differential equations, Itogi Nauki i Tekh. Ser. Sovrem. Probl. Mat. 30 (1987), 105-201 (in Russian); English transl.: J. Soviet Math. 43 (1988), 2340-2417.
• [12] T. Kusano and M. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81-99.
• [13] T. Kusano, Y. Naito and A. Otaga, Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations Dynam. Systems 2 (1994), 1-10.
• [14] H. J. Li, Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J. 25 (1995), 571-583.
• [15] H. J. Li and C. C. Yeh, Sturmian comparison theorem for half-linear second order differential equations, Proc. Roy. Soc. Edinburgh 125A (1996), 1193-1204.
• [16] A. Lomtatidze, Oscillation and nonoscillation of Emden-Fowler type equation of second order, Arch. Math. (Brno) 32 (1996), 181-193.
• [17] J. D. Mirzov, On some analogs of Sturm's and Kneser's theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418-425.
• [18] J. D. Mirzov, Asymptotic Properties of Nonlinear Nonautonomous Systems of Ordinary Differential Equations, Adygeja, Majkop, 1994.
• [19] Z. Opial, Sur une inégalité de Ch. de la Vallée Poussin dans la théorie de l'équation différentielle linéaire du second ordre, Ann. Polon. Math. 6 (1959), 81-91.
• [20] G. Sansone, Equazioni differenziali nel campo reale, parte prima, Zanichelli, Bologna, 1948.
• [21] Ch. de la Vallée Poussin, Sur l'équation différentielle linéaire du second ordre. Détermination d'une intégrale par deux valeurs assignés. Extension aux équations d'ordre n, J. Math. Pures Appl. 8 (1929), 125-144.
• [22] D. Willet, Generalized de la Vallée Poussin disconjugacy test for linear differential equations, Canad. Math. Bull. 14 (1971), 419-428.